Coordinated flow pipe elbow

A pipe elbow having an interior surface, an exterior surface, an interior, a centerline, and a bend having an axis of rotation and an apex, where the interior surface of the bend is defined by an inside curve and an outside curve, and intrados and extrados at a given cross-section of the bend, said pipe elbow comprising:

• an inlet portion having an interior cross-section with a first cross-sectional configuration and a first cross-sectional area, wherein said first cross-section configuration is circular;
• an outlet portion having an interior cross-section with a second cross-sectional configuration and second cross-sectional area, wherein said second cross-section configuration is circular;
• and
• an intermediate portion joined to said inlet and outlet portions between said inlet portion and said outlet portion, wherein said intermediate portion forms said bend, and said intermediate portion has an interior cross-section with a third cross-sectional configuration and a third cross-sectional area, wherein the third cross-sectional configuration of the intermediate portion is non-circular, wherein:
• the interior cross-section of said intermediate portion is in the configuration of a closed non-circular curvilinear figure which has a first region between the centerline and the intrados and a second region between the centerline and the extrados, wherein the first region has a larger cross-sectional area than said second region,
• said closed non-circular curvilinear figure has a round configuration that is asymmetrical about the centerline so that the interior of the intermediate portion is configured to provide a greater volume of flow at the inside of the bend and less volume at the outside of the bend,
• said closed non-circular curvilinear figure has a height which is greatest between said intrados and said centerline, and
• said closed non-circular curvilinear figure comprising three convexly-curved portions with a smaller radius of curvature, alternating with three convexly-curved portions with a larger radius of curvature, wherein the entire interior cross-section of the intermediate portion is comprised of convex curvilinear segments, and said convexly-curved portions are arranged as follows:
• wherein one of said convexly-curved portions with a smaller radius of curvature is located at the extrados;
• two of said convexly-curved portions with a smaller radius of curvature are located between said intrados and said centerline;
• and
• one of said convexly-curved portions with a larger radius of curvature is located adjacent said intrados and extends from one of said two convexly-curved portions with a smaller radius of curvature that is located between said intrados and said centerline to the other convexly-curved portion with a smaller radius of curvature that is located between said intrados and said centerline,
• wherein the first, second, and third cross-sectional areas are the same, and the bend of said pipe elbow has a variable radius of curvature, and
• the configuration of said intermediate portion provides an increased volume for fluid to flow through the inside of the bend and less volume for fluid to flow at the outside of the bend.

A method of designing a pipe elbow having a centerline and a bend therein, wherein the pipe elbow has an inlet portion, an outlet portion, and an intermediate portion joined to said inlet and outlet portions between the inlet portion and the outlet portion, wherein the intermediate portion forms the bend, and the method comprises performing the following steps in any order:

• 1) Designing an inlet portion having an interior cross-section with a first cross-sectional configuration and a first cross-sectional area;
• 2) designing an outlet portion having an interior cross-section with a second cross-sectional configuration and second cross-sectional area;
• and
• 3) Designing an intermediate portion which has an interior cross-section with a third cross-sectional configuration and a third cross-sectional area, wherein the third cross-sectional configuration of the intermediate portion is different from the first and second cross-sectional configuration, wherein:
• a) the intermediate portion has interior cross-sections in configurations that vary along the centerline and are defined by the equation:
• a
• =
• {
• R
• ⁡
• (
• ξ
• )
• ρ
• ⁢
• r
• 2
• -
• (
• ρ
• -
• R
• ⁡
• (
• ξ
• )
• )
• 2
• -
• R
• ⁡
• (
• ξ
• )
• ρ
• ⁢
• r
• 2
• -
• (
• ρ
• -
• R
• ⁡
• (
• ξ
• )
• )
• 2
• where:
• ρ is the radial coordinate, R is the radius of curvature, e is Euler's number, r is the radius of a circular inlet, and ξ is a variable indicating progress along the centerline path parametrization variable, where ξ=0 at the apex of the bend;
• b) the interior of the intermediate portion is oriented to provide a greater volume at the inside of the bend and less volume at the outside of the bend according to the equation in a) above which equation specifies how volume is reallocated by increasing the height out-of-plane of the bend plane in proportion to the inverse of how far that cross-section of the elbow is away from the center of the bend;
• c) there is a smooth transition between the interior cross-section of the intermediate portion and the first cross-sectional configuration of the inlet and the second cross-sectional configuration of the outlet;
• and
• d) the centerline turns mathematically smoothly as defined by a bump function with compact support, wherein a bump function is a curve that is infinitely differentiable, and compact support means that the function is zero outside of a finite range.